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Number Theory in 3d Gravity and from 4d Gauge Theory


String theory is very successful in connecting deep results between physics and mathematics. However, besides the appearance of modular forms in many supersymmetric partition functions, the relations between string theory and number theory are relatively weak, compared with geometry, algebraic/geometric topology and representation theory. In this thesis, we aim to strengthen the tie between high-energy physics and various aspects of number theory. We start by investigating Euclidean AdS$_3$ quantum gravity, with asymptotic boundary as either genus-one, namely torus, or higher-genus Riemann surfaces. We will see the appearance of the asymptotic analysis on coefficients in the Klein’s j-function in Chapter 2, and two exotic finite-index subgroups of $SL(2,\mathbb{Z})$, with indices 24 and 384 respectively, in Chapter 3. In the end of that chapter, we also conjecture the existence of a series of finite-index subgroups for conformal boundary of any genus. Then we switch gear to the realm of $p$-adic numbers, and study the discrete semiclassical gravity on a Bruhat-Tits tree and its quotient, where we encounter a special representation of $PGL(2,\mathbb{Q}_p)$ and we explore its surprising relation to Chebyshev polynomials. At this point, we will have observed fascinating number-theoretic objects in studying 3d gravity and 2d conformal field theory. Finally, we present an example in which we are able to derive sophisticated number-theoretic identities, including the classic quadratic reciprocity by Gauss, from a careful use of first principles in string theory and 4d supersymmetric gauge theories.

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